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1 . 在世界环保意识日益强化,石油资源日渐沽竭的今天,以氢气做动力源的研究已成为一大课题.当年马自达坚持下来的转子发动机(如图1)从结构上讲是最适合燃烧氢气,而且最“干净”,因为氢燃烧完后排出的是水蒸气,对环境没有任何污染.马自达公司改制了RX-7型跑车的转子发动机,使它可以用氢做燃料.以正四面体的四个顶点为球心,以正四面体的棱长为半径的四个球的相交部分围成的几何体(如图2)被称为“勒洛四面体”,它表面上任意两点间的距离最大值与正四面体棱长相等,能在两个平行平面间自由转动,并且始终保持与两平面都接触,因此它能像球一样来回滚动.转子发动机的设计正是利用了这一原理.转子引擎只需转一周,各转子便有一次进气、压缩、点火与排气过程,相当于往复式引擎运转两周,因此具有小排气量就能成就高动力输出的优点.另外,由于转子引擎的轴向运动特性,它不需要精密的曲轴平衡就可以达到非常高的运转转速.若正四面体ABCD的棱长为2,将对应的勒洛四面体ABCD放进一个正方体纸盒中,若该勒洛四面体可以在纸盒内任意转动,则该纸盒棱长的最小值为__________ ;若在勒洛四面体ABCD内放一个小正方体零件,该零件可以在勒洛四面体ABCD内任意转动,则该零件棱长的最大值为__________ .
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2卷引用:山西省介休市第一中学校2022-2023学年高二上学期开学考试数学试题
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2 . 在
中,角
的对边分别为
,
,
,且
,则
的形状为_________ 三角形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/323b8b64ccdb829eaefa2180796012eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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3 . 已知
.
(1)若
,求
;
(2)若
互斥,求
;
(3)若
相互独立,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e5c94d92ea7cefd91905c9ae6147f6.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf22d7d1a965bda25168a233fb6290c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b16555b699d84c28e8e2dcd66e382cc9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b16555b699d84c28e8e2dcd66e382cc9.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b16555b699d84c28e8e2dcd66e382cc9.png)
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4 . 已知正三棱锥
的顶点为
,底面是正三角形
.
两两所成角为
,设质点
自
出发,依次沿着三个侧面移动环绕一周,直至回到出发点
,求质点移动路程的最小值;
(2)若该三棱锥的所有棱长均为1,求以
为顶点,以三角形
内切圆为底面的圆锥的侧面积;
(3)若该三棱锥的体积为定值
,求该三棱锥侧面与底面所成的角
的正切值,使该三棱锥的表面积
最小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19bb1063e139610045f3bca5ca0b2766.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea31f8a526b3d83b099f43086ba950d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若该三棱锥的所有棱长均为1,求以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(3)若该三棱锥的体积为定值
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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5 . 从3,4,5,6四个数中任取两个数,则两个数之差为2的概率是______ .
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6 . 设
(其中
为虚数单位),则
的虚部为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb8ac4453aa8b5c00111ad671e443476.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
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7 . 已知梯形
,
,
,
,
,
是线段
的中点.将
沿着
所在的直线翻折成四面体
,翻折的过程中下列选项正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d262480ffb55b7617f44b63f130c154a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f571396be1aa4a8914a66f7d7abd6381.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9060f03b9ee41d70d135b1e1a8902ce9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.![]() ![]() |
B.当直线![]() ![]() ![]() ![]() |
C.四面体![]() ![]() |
D.四面体![]() ![]() |
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8 . 已知
,
,且
,
均为锐角,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a077ceccea768ed3c664d38d55242fbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06397fbd8b97298fc8bf8f1f42e21d92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eacde1c42151734fdc60f3001b590de.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
9 . 如图,在正方体
中,
,
分别为
,
的中点,则异面直线
与
所成角的大小为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c09eec4e14a861af83d7828797d176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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10 . 已知
为虚数单位,
,复数
的共轭复数为
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d971e248bd035650f7fc77adaea2ba1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fcfebd9f5a57036e6df6b6e14865da3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa63536f51438bae1a5c3d6d2bd2d01.png)
A.0 | B.1 | C.2 | D.![]() |
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